Studies On Stirling Sums And Euler Sums

The Euler sums are infinite series involving harmonic numbers.This kind of series has received wide concern.However,there are still a large number of Euler sums' values that have not been determined.In this paper,using the Bell polynomials,generating functions,multiple zeta values and integrals of special functions,we establish various mixed Euler sums and Stirling sums,and present a unified approach to determining the evaluations of all the unknown Euler sums of weights {6,7,8.9,10?.As a result.we give the values of two Euler sums of weight 6,six sums of weight 7,twelve sums of weight 8,sixteen sums of weight 9 and twenty-seven sums of weight 10.The main contents are as follows:(1)By the special integrals associated with logarithmic function and polylogarithm functions,we establish three forms of infinite series involving the unsigned Stirling numbers of the first kind,including the Stirling sums related to the integrals W(k,m),the Stirling sums containing the sequence Yk(n)and the Stirling sums with the integrals Vn(p,q).Moreover,we introduce some other known Stirling sums which are utilized in this thesis to compute the unknown Euler sums.(2)By the integrals of special functions,we also obtain the generating functions of several sequences involving harmonic numbers,and establish some relations,from which,we construct the corresponding mixed Euler sums.In addition,we obtain three relations between the multiple zeta values and Euler sums,and calculate some special multiple zeta values.(3)We put,forward a unified method for computing the values of all the unknown Euler sums of weights {6,7,8,9,10} by using Stirling sums,mixed Euler sums and Bell polynomials,and arrival at a conclusion that all the Euler sums of weights {6,7,9} are reducible to zeta values,and all the Euler sums of weight 8 are expressible in terms of linear sum S2,6 and zeta values,and all the Euler sums of weight 10 can reduce to linear sums S2,6,S2,8 and zeta values.